A method and device for identifying a structural system during the cantilever casting construction of an arch bridge

Abstract

The present invention belongs to the technical field related to health detection, and discloses a method and device for identifying a structural system during the cantilever casting construction of an arch bridge. The method includes the following steps: dividing the construction process of the arch bridge cantilever into several construction stages, designing the measuring point positions based on the observable state equations corresponding to the construction stages, and then using the observability technology to identify the structural system for each construction stage one by one, so as to obtain the flexural stiffness and axial stiffness of all arch rib beam elements in the corresponding construction stage. In each construction stage of the cantilever casting of the arch bridge, the present invention transforms the equivalent stiffness matrix equation into an observable state equation, and uses the null space method to solve the observable state equation, obtaining the analytical solutions of the axial stiffness and bending stiffness of the arch rib beam segment elements in each construction stage, and improving the identification accuracy.

Description

Technical Field

[0001] The present invention belongs to the technical field related to health detection, and more specifically, relates to a method and device for identifying a structural system during the cantilever casting construction of an arch bridge. Background Art

[0002] Due to the continuous progress of construction technology, reinforced concrete arch bridges have become increasingly common in the field of long-span bridges. The cantilever casting method has been widely used in bridges with spans ranging from 150 meters to 300 meters in recent years. This method is widely used because of its strong applicability, high safety, reasonable cost, and developed technology.

[0003] Currently, the focus of stress research during the construction of the main arch ring is to ensure the safety of the main arch ring by optimizing the initial tension of the stay cables. However, in practical applications, the method of adjusting the arch ring section stress through the stay cables has certain limitations. The adjustable range is limited, and due to the uncertainty of temporary loads and the control accuracy of on-site tension, there is a difference between the actual stress of the arch ring and the theoretically calculated stress. Summary of the Invention

[0004] In view of the above defects or improvement requirements of the prior art, the present invention provides a method and device for identifying a structural system during the cantilever casting construction of an arch bridge to achieve parameter update and state prediction of the arch rib segment stiffness with the construction process.

[0005] To achieve the above object, according to one aspect of the present invention, a method for identifying a structural system during the cantilever casting construction of an arch bridge is provided. The method includes the following steps:

[0006] Divide the construction process of the arch bridge cantilever into several construction stages, design the measuring point positions based on the observable state equations of the corresponding construction stages, and then use the observability technology to identify the structural system for each construction stage one by one to obtain the flexural stiffness and axial stiffness of all arch rib beam elements in the corresponding construction stage;

[0007] Among them, the identification steps of the flexural stiffness and axial stiffness of the beam elements in the first construction stage are as follows: First, establish the equilibrium equation of the first construction stage of the arch bridge cantilever based on the stiffness method, and then obtain the equivalent stiffness matrix equation of the beam segment in the first construction stage;

[0008] Then, introduce the boundary conditions, use the observability technology to separate the unknowns and knowns in the equivalent stiffness matrix equation of the beam segment in the first construction stage, and then construct the dynamic observability state equation, where the knowns include the rotation angles of the nodes;

[0009] Finally, use the null space matrix of the coefficient matrix of the dynamic observability state equation to determine the solution of the dynamic observability state equation, and then obtain the axial stiffness and flexural stiffness of the arch rib beam elements in the first construction stage.

[0010] Furthermore, the matrix form of the equilibrium equation in the first construction stage is as follows:

[0011]

[0012] In the formula: [K] is the stiffness matrix, corresponding to the characteristics of two-dimensional beam elements, including the element length L i , cross-sectional area A i , moment of inertia of the cross-section I zi and elastic modulus E i , i is the unit number; {δ} is the displacement column vector, including the horizontal displacement u j , vertical displacement v j and rotation angle w j ; {f} is the force column vector, including axial force H j , vertical force V j , bending moment M j , j is the node number.

[0013] Furthermore, the equivalent stiffness matrix equation of the beam segment in the first construction stage is:

[0014]

[0015] where H, V, and M are force vectors, and u, v, and w are the displacements and rotations of each node; E1A1 is the axial stiffness of the beam element in the first construction stage, and E1I1 is the flexural stiffness of the beam element in the first construction stage.

[0016] Furthermore, the dynamic observability state equation corresponding to the first construction stage is:

[0017]

[0018] Furthermore, solve the null space matrix V of the coefficient matrix B. If V does not exist, that is, when V is zero-dimensional, there is a unique solution for all variables in the equation; when V exists and is not O, the variables corresponding to all zero rows of it have a unique solution, and the corresponding unique particular solution variables are obtained through B\D, and then the axial stiffness and flexural stiffness of the arch rib beam element in the first construction stage are obtained.

[0019] Furthermore, compare the identified stiffness parameters of the beam segment with the corresponding original beam segment design stiffness parameters. When the obtained difference is greater than the set threshold, the dynamic observability state equation adopts the identified stiffness parameters.

[0020] Furthermore, obtain the coefficient matrix B of the corrected dynamic observability state equation, and perform inverse analysis using the observability state equation of the structure to obtain the deflections and rotations at each node.

[0021] The present invention also provides a structural system identification system during the cantilever casting construction of an arch bridge. The system includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it executes the structural system identification method during the cantilever casting construction of the arch bridge as described above.

[0022] The present invention also provides a computer-readable storage medium. The computer-readable storage medium stores machine-executable instructions. When the machine-executable instructions are called and executed by a processor, the machine-executable instructions prompt the processor to implement the structural system identification method during the cantilever casting construction of the arch bridge as described above.

[0023] Generally speaking, compared with the prior art through the above technical solutions conceived by the present invention, the structural system identification method and device provided by the present invention during the cantilever casting construction of the arch bridge mainly have the following beneficial effects:

[0024] 1. At each construction stage of the cantilever casting of the arch bridge, the equivalent stiffness matrix equation is transformed into an observable state equation, and the null space method is used to solve the observable state equation, obtaining the analytical solutions of the axial stiffness and bending stiffness of each arch rib beam segment unit. Through verification with the finite element model, the results show that the accuracy of the identification of the structural axial and bending stiffness by this method is very high.

[0025] 2. As the number of segments increases, the axial stiffness (EA) is precisely controlled through observability technology and within the calculation error range. However, the error of the bending stiffness (EI) is unacceptable. To solve this problem, the rotation angle of the corresponding node is also taken as a measurable parameter, and then the axial stiffness and bending stiffness calculated by using the observability technology are in good agreement with the actual situation.

[0026] 3. During the construction process, the stiffness of the arch rib beam unit is not constant. Ignoring this factor will have an adverse impact on the response prediction of the arch rib beam unit. To solve this problem, the change of the stiffness parameter is updated in real time to predict the actual displacement of each unit. Specifically, the inverse analysis of the structural equation during the construction stage is carried out by using the observability technology: first, the error between the identified stiffness parameter of the beam segment and the corresponding original beam segment design stiffness parameter is compared to select the one with the smaller corresponding error. The dynamic observability state equation adopts the stiffness parameter corresponding to the one with the smaller error, and then the coefficient matrix B of the corrected dynamic observability state equation is obtained. The inverse analysis is carried out by using the dynamic observability state equation of the structure to obtain the deflection and rotation angle at each node. The results show that these observation results are very close to the actual deflection and rotation angle obtained from the analysis of the updated finite element model. Description of the Drawings

[0027] Figure 1 It is a schematic diagram of the first construction stage;

[0028] Figure 2 It is a schematic diagram of the second construction stage;

[0029] Figure 3 It is a schematic diagram of the flexural rigidity correction of the arch rib part;

[0030] Figure 4 It is a flowchart related to a structural system identification method during the cantilever casting construction of an arch bridge provided by the present invention. Specific embodiments

[0031] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0032] Please refer to Figure 4 , the present invention provides a structural system identification method during the cantilever casting construction of an arch bridge. The method includes the following steps: dividing the construction process of the arch bridge cantilever into several construction stages, designing the measuring point positions based on the observable state equations of the corresponding construction stages, and then using the observability technology to identify the structural system for each construction stage one by one to obtain the flexural rigidity and axial rigidity of all arch rib beam elements in the corresponding construction stage;

[0033] Please refer to Figure 1 , Figure 2 and Figure 3 , wherein, the identification steps of the flexural rigidity and axial rigidity of the beam elements in the first construction stage are as follows: first, establish the equilibrium equation of the first construction stage of the arch bridge cantilever based on the stiffness method, and then use the observability technology to identify the structural system of the first construction stage to obtain the flexural rigidity and axial rigidity of the corresponding arch rib beam elements.

[0034] It can be known from the stiffness method that for a two-dimensional structure with N N nodes, the matrix form of its equilibrium equation is:

[0035]

[0036] In the formula: [K] is the stiffness matrix, corresponding to the characteristics of the two-dimensional beam element, including the element length L i , cross-sectional area A i , moment of inertia of the cross-section I zi and elastic modulus E i (i is the unit number); {δ} is the displacement column vector, including the horizontal displacement uj , the vertical displacement v j and the rotation angle w j (j is the node number); {f} is the force column vector, including the axial force H j , the vertical force V j , the bending moment M j , j is the node number.

[0037] There are 2 nodes in the beam element of the first construction stage, and the external loads at its nodes include the weight of the hanging basket and the self-weight of the newly cast section. Assuming the weight of the hanging basket is 900 kN and the weight of the newly cast section is 2150 kN, and the characteristics of the beam segment unit are all known. Thus, the equivalent stiffness matrix equation of the beam segment in the first construction stage can be obtained as follows:

[0038]

[0039] where H, V, and M are force vectors, and u, v, and w are the displacements and rotations of each node; E1A1 is the axial stiffness of the beam element in the first construction stage, and E1I1 is the flexural stiffness of the beam element in the first construction stage. The subscript represents the node number of the beam segment in the first construction stage.

[0040] From the boundary conditions, it can be known that u1, v1, w1 = 0 in the above formula, and u2 and v2 can be measured, so the above quantities are all known quantities.

[0041] Using the observability technology to separate the unknowns and knowns in the equivalent stiffness matrix equation of the beam segment in the construction stage, and then constructing the dynamic observability state equation, where the knowns include the rotations of the nodes, and the dynamic observability state equation is:

[0042]

[0043] Finally, use the null space matrix of the coefficient matrix of the dynamic observability state equation to judge the solution of the dynamic observability state equation, and then obtain the axial stiffness and flexural stiffness of the arch rib beam element in the first construction stage.

[0044] Specifically, perform observability analysis based on the null space matrix. Solve the null space matrix V of the coefficient matrix B. If V does not exist, that is, when V is zero-dimensional, there is a unique solution for all variables in the equation; when V exists and is not O, the variables corresponding to the all-zero rows of it have a unique solution, and the corresponding unique particular solution variables are obtained through B\D. Thus, the flexural stiffness and axial stiffness of the first segment of the beam can be obtained as:

[0045]

[0046] Use the identification method in the first construction stage to identify the flexural stiffness and axial stiffness of the beam element in the second construction stage, so as to obtain the flexural stiffness and axial stiffness of the corresponding arch rib beam element.

[0047] The beam elements in the second construction stage have a total of 3 nodes, and the external loads on the nodes include the weight of the hanging basket and the self-weight of the newly cast section. The weight of the hanging basket is 900 kN, and the weight of the newly cast section is 2150 kN.

[0048] Thus, the equivalent stiffness matrix equation for the first two sections of the beam is as follows:

[0049]

[0050] Where H, V, and M are force vectors, u, v, and w are the displacements and rotations of each node, and they need to correspond to the symbols in the formula. K a , K c , and K b are the stiffness matrices of the first section of the beam, and K d , K e , and K f are the stiffness matrices of the second section of the beam.

[0051] From the boundary conditions, it can be seen that in the above formula, u1, v1, w1 = 0, and u2, v2 and u3, v3 can be measured, so the above quantities are all known quantities.

[0052] Similarly, according to the observability method, separate variables and transform it into the standard form of the observability equation:

[0053]

[0054] Finally, perform observability analysis based on the null space matrix. Solve the null space matrix V of the coefficient matrix B. If V does not exist, that is, when V is zero-dimensional, there is a unique solution for all variables in the equation; when V exists and is not O, the variables corresponding to the all-zero rows of it have a unique solution, and the corresponding unique particular solution variables are obtained through B\D. Thus, the solution of the observability state equation for this construction stage is as follows:

[0055]

[0056] Use the observability technology to identify the flexural stiffness and axial stiffness of the arch rib beam elements corresponding to the subsequent construction stages.

[0057] Based on the above observability techniques, the axial stiffness and flexural stiffness of each construction stage can be identified. However, as the number of segments increases, the axial stiffness (EA) is precisely controlled by the observability technique and within the range of computational error. However, the error of the flexural stiffness (EI) is unacceptable. To solve this problem, the rotation angle of the corresponding node is also taken as a measurable parameter, and then the axial stiffness and flexural stiffness calculated by the observability technique are in good agreement with the actual situation. During the construction process, the stiffness of the arch rib beam element is not constant. Ignoring this factor will have an adverse impact on the response prediction of the arch rib beam element. To solve this problem, the change of the stiffness parameter is updated in real time to predict the actual displacement of each element. Specifically, the inverse analysis of the structural equation of the construction stage is carried out using the observability technique: the identified stiffness parameter of the beam segment is compared with the corresponding original beam segment design stiffness parameter. When the obtained difference is greater than the set threshold, the dynamic observability state equation adopts the identified stiffness parameter, and then the coefficient matrix B of the modified dynamic observability state equation is obtained. The inverse analysis is carried out using the dynamic observability state equation of the structure to obtain the deflection and rotation angle at each node. The results show that these observation results are very close to the actual deflection and rotation angle obtained from the analysis of the updated finite element model.

[0058] This method ensures that the model is dynamically consistent with the changing physical conditions on site, providing more accurate and realistic structural response predictions during construction.

[0059] The following is a further detailed description of the present invention in specific embodiments.

[0060] This specific embodiment includes the following steps:

[0061] 1) Divide the construction process of the arch bridge into several construction stages, and design the measuring point positions based on the observability state equation in the corresponding construction stage. Use the observability technique to identify the structural system for each stage one by one, and obtain the flexural stiffness and axial stiffness of all arch rib beam elements in the corresponding construction stage.

[0062] S1: First, establish the equilibrium equation of the structure based on the stiffness method, and then use the observability technique to identify the structural system of the first construction stage to obtain the flexural stiffness and axial stiffness of the corresponding arch rib beam elements.

[0063] According to the stiffness method, for a two-dimensional structure with N N nodes, the matrix form of its equilibrium equation is:

[0064]

[0065] In the formula: [K] is the stiffness matrix, corresponding to the characteristics of the two-dimensional beam element, including the element length L i, cross-sectional area A i , moment of inertia of cross-section I zi and elastic modulus E i (i is the unit number); {δ} is the displacement column vector, including the horizontal displacement u j , vertical displacement v j and rotation angle w j (j is the node number); {f} is the force column vector, including the axial force H j , vertical force V j , bending moment M j (j is the node number).

[0066] The beam element in the first construction stage has 2 nodes in total, and the external loads at its nodes include the weight of the hanging basket and the self-weight of the newly cast section. Assuming the weight of the hanging basket is 900 kN and the weight of the newly cast section is 2150 kN, and the characteristics of the beam section elements are all known. Thus, the equivalent stiffness matrix equation of the first section of the beam can be obtained as follows:

[0067]

[0068] where H, V, and M are the force vectors, and u, v, and w are the displacements and rotations of each node. EA is the axial stiffness of the beam element, and EI is the flexural stiffness of the beam element. The subscript 1 of EA and EI represents the beam section in the first construction stage.

[0069] From the boundary conditions, it can be known that u1, v1, w1 = 0 in the above formula, and u2 and v2 can be measured, so all the above quantities are known quantities.

[0070] Using the observability technology to separate the unknown and known quantities in the equation, and constructing the standard form B·z = D of the dynamic observability state equation as follows:

[0071]

[0072] Finally, based on the null space matrix, the observability analysis is carried out. Solve the null space matrix V of the coefficient matrix B. If V does not exist, that is, when V is zero-dimensional, there is a unique solution for all variables in the equation; when V exists and is not O, the variables corresponding to the all-zero rows of it have a unique solution, and the corresponding unique particular solution variables can be obtained through B\D. Thus, the flexural stiffness and axial stiffness of the first section of the beam can be obtained as follows:

[0073]

[0074] S2: Then, according to the method in step S1, the structural system identification of the second construction stage is carried out to obtain the flexural stiffness and axial stiffness of the corresponding arch rib beam element.

[0075] The beam elements in the second construction stage have a total of 3 nodes, and the external loads at the nodes include the weight of the hanging basket and the self-weight of the newly cast section. The weight of the hanging basket is 900 kN, and the weight of the newly cast section is 2150 kN.

[0076] Thus, the equivalent stiffness matrix equations for the first two segments of the beam are as follows:

[0077]

[0078] Where H, V, and M are force vectors, u, v, and w are the displacements and rotations of each node, and K a , K c , and K b are the stiffness matrices of the first segment of the beam, and K d , K e , and K f are the stiffness matrices of the second segment of the beam.

[0079] From the boundary conditions, it can be seen that in the above equation, u1, v1, w1 = 0, and u2, v2, and u3, v3 can be measured, so the above quantities are all known quantities.

[0080] Similarly, according to the observability method, separate variables and transform it into the standard form of the observability equation:

[0081]

[0082] Finally, perform observability analysis based on the null space matrix. Solve the null space matrix V of the coefficient matrix B. If V does not exist, that is, when V is zero-dimensional, there is a unique solution for all variables in the equation; when V exists and is not O, the variables corresponding to all its zero rows have a unique solution, and the corresponding unique particular solution variables are obtained through B\D. Thus, the solution of the observability state equation for this construction stage is as follows:

[0083]

[0084] S3: Use the observability technology to identify the flexural stiffness and axial stiffness of the arch rib beam elements corresponding to the subsequent construction stages.

[0085] As described in sub-steps S1 and S2 above, the axial and flexural stiffnesses of each construction stage can be identified based on the above observability technology. However, as the number of segments increases, the axial stiffness (EA) is precisely controlled by the observability technology and is within the calculation error range. However, the error of the flexural stiffness (EI) is unacceptable. To solve this problem, we also take the rotation angle of the corresponding node as a measurable parameter, and then the axial stiffness and flexural stiffness calculated using the observability technology are in good agreement with the actual situation.

[0086] 2) Structural response prediction

[0087] During the construction process, the stiffness of the arch rib beam element is not constant. Ignoring this factor will have an adverse impact on the response prediction of the arch rib beam element. To solve this problem, the changes in the stiffness parameters in the finite element model are updated in real time to predict the actual displacement of each element. Using the observability technique, the system identification process will first evaluate the updated bending stiffness parameters within the design parameters. Subsequently, using the updated stiffness parameters, the vertical deflection of each new segment will be predicted at each construction stage. This method ensures that the model is dynamically consistent with the changing physical conditions on site, providing a more accurate and realistic prediction of the structural response during construction.

[0088] During the construction of the arch bridge, monitoring data such as displacement and rotation angle are used in the structural system identification technology. This technology is used to identify, update, and establish the structural physical parameters of the current construction stage. The updated structural state inverse equation predicts the actual displacement and rotation angle of the structure at the next construction stage. And it is compared with the actual monitoring results to verify the reliability of the structural system identification algorithm during the construction of the cantilever arch bridge.

[0089] The main difference between this step and the traditional method is that applying the structural system identification and observability techniques to the construction analysis of the cantilever-cast arch bridge can more accurately identify the axial stiffness and bending stiffness of the beam segments during the cantilever-cast construction of the arch bridge, and can predict the structural displacement at the next construction stage.

[0090] The present invention also provides a structural system identification system during the cantilever-cast construction of an arch bridge. The system includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, it executes the structural system identification method during the cantilever-cast construction of an arch bridge as described above.

[0091] The present invention also provides a computer-readable storage medium. The computer-readable storage medium stores machine-executable instructions. When the machine-executable instructions are called and executed by the processor, the machine-executable instructions cause the processor to implement the structural system identification method during the cantilever-cast construction of an arch bridge as described above.

[0092] Those skilled in the art can easily understand that the above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for identifying the structural system during the cantilever casting construction of an arch bridge, characterized in that, The method comprises the following steps: The construction process of the arch bridge cantilever is divided into several construction stages. Based on the observable state equations of the corresponding construction stages, the measuring point positions are designed, and then the structural system identification is carried out for each construction stage one by one by using the observability technology to obtain the flexural stiffness and axial stiffness of all arch rib beam elements in the corresponding construction stage; Among them, the identification steps of the flexural stiffness and axial stiffness of the beam elements in the first construction stage are as follows: First, based on the stiffness method, the equilibrium equation of the first construction stage of the arch bridge cantilever is established, and then the equivalent stiffness matrix equation of the beam segment in the first construction stage is obtained; Then, boundary conditions are introduced, and the observability technology is used to separate the unknowns and knowns in the equivalent stiffness matrix equation of the beam segment in the first construction stage, and then a dynamic observability state equation is constructed, where the knowns include the rotation angles of the nodes; Finally, the null space matrix of the coefficient matrix of the dynamic observability state equation is used to judge the solution of the dynamic observability state equation, and then the axial stiffness and flexural stiffness of the arch rib beam elements in the first construction stage are obtained.

2. The structural system identification method during the cantilever casting construction of an arch bridge as described in claim 1, wherein: The matrix form of the equilibrium equation in the first construction stage is: Where: [K] is the stiffness matrix, corresponding to the characteristics of a two-dimensional beam element, including the element length L i , the cross-sectional area A i , the moment of inertia of the cross-section I zi and the elastic modulus E i , i is the unit number; {δ} is the displacement column vector, including the horizontal displacement u j , the vertical displacement v j and the rotation angle w j ; {f} is the force column vector, including the axial force H j , the vertical force V j , the bending moment M j , j is the node number.

3. The structural system identification method during the cantilever casting construction of an arch bridge according to claim 2, wherein: The equivalent stiffness matrix equation of the beam segment in the first construction stage is: Where H, V, and M are force vectors, and u, v, and w are the displacements and rotation angles of each node; E1A1 is the axial stiffness of the beam element in the first construction stage, and E1I1 is the flexural stiffness of the beam element in the first construction stage.

4. The structural system identification method during the cantilever casting construction of an arch bridge according to claim 3, characterized in that: The corresponding dynamic observability state equation in the first construction stage is:

5. The structural system identification method during the cantilever casting construction of an arch bridge according to claim 4, characterized in that: Solve the null space matrix V of the coefficient matrix B. If V does not exist, that is, when V is zero-dimensional, there is a unique solution for all variables in the equation; when V exists and is not O, there is a unique solution for the variables corresponding to all zero rows of it. The corresponding unique particular solution variables are obtained through B\D, and then the axial stiffness and flexural stiffness of the arch rib beam elements in the first construction stage are obtained.

6. The structural system identification method during the cantilever casting construction of an arch bridge according to any one of claims 1-5, characterized in that: Compare the identified stiffness parameters of the beam segment with the corresponding original beam segment design stiffness parameters. When the obtained difference is greater than the set threshold, the dynamic observability state equation adopts the identified stiffness parameters.

7. The structural system identification method during the cantilever casting construction of an arch bridge according to claim 6, wherein: Obtain the coefficient matrix B of the corrected dynamic observability state equation, and use the observability state equation of the structure to perform inverse analysis to predict the deflections and rotation angles at the corresponding nodes in the next construction stage.

8. A structural system identification system during the cantilever casting construction of an arch bridge, characterized in that: The system comprises a memory and a processor. The memory stores a computer program. When the processor executes the computer program, it executes the structural system identification method during the casting construction of the arch bridge cantilever according to any one of claims 1-7.

9. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores machine-executable instructions. When the machine-executable instructions are called and executed by the processor, the machine-executable instructions cause the processor to implement the structural system identification method during the casting construction of the arch bridge cantilever according to any one of claims 1-7.

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